#3: Derivatives of Other Functions

Question 1

What is the derivative of the exponential function f(x) = e^x?

Answer 1

The derivative is f’(x) = e^x.

Question 2

True or False: The derivative of f(x) = a^x (where a > 0) is f’(x) = a^x ln(a).

Answer 2

True.

Question 3

Fill in the blank: The derivative of the natural logarithm function f(x) = ln(x) is _____.

Answer 3

f’(x) = 1/x.

Question 4

What is the derivative of the logarithmic function f(x) = log_a(x)?

Answer 4

The derivative is f’(x) = 1/(x ln(a)).

Question 5

Multiple Choice: What is the derivative of f(x) = 2^x?

Answer 5

f’(x) = 2^x ln(2).

Question 6

What is the power rule for differentiation?

Answer 6

If f(x) = x^n, then f’(x) = n*x^(n-1).

Question 7

True or False: The product rule states that if u(x) and v(x) are differentiable functions, then (u*v)’ = u’v + uv’.

Answer 7

True

Question 8

Fill in the blank: The quotient rule states that if f(x) = u(x)/v(x), then f’(x) = ________.

Answer 8

(u’v - uv’) / v^2

Question 9

What is the derivative of the inverse function f^{-1}(x) at a point a?

Answer 9

If f’(f^{-1}(a)) ≠ 0, then
(f^{-1})’(a) = 1 / f’(f^{-1}(a)).

Question 10

Multiple Choice: Which of the following is the correct application of the product rule? If f(x) = u(x)v(x), what is f’(x)?
A) u’v
B) u’v + uv’
C) u’v’
D) uv

Answer 10

B) u’v + uv’